## Book description

Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas.

Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented.

Principles of Fourier Analysis stimulates the appreciation and understanding of the fundamental concepts and serves both beginning students who have seen little or no Fourier analysis as well as the more advanced students who need a deeper understanding. Insightful, non-rigorous derivations motivate much of the material, and thought-provoking examples illustrate what can go wrong when formulas are misused. With clear, engaging exposition, readers develop the ability to intelligently handle the more sophisticated mathematics that Fourier analysis ultimately requires.

1. Cover
2. Half Title
3. Title Page
6. Preface
7. Sample Courses
8. I Preliminaries
1. 1 The Starting Point
2. 2 Basic Terminology, Notation and Conventions
3. 3 Basic Analysis I: Continuity and Smoothness
4. 4 Basic Analysis II: Integration and Infinite Series
5. 5 Symmetry and Periodicity
6. 6 Elementary Complex Analysis
7. 7 Functions of Several Variables
9. II Fourier Series
1. 8 Heuristic Derivation of the Fourier Series Formulas
2. 9 The Trigonometric Fourier Series
3. 10 Fourier Series over Finite Intervals (Sine and Cosine Series)
4. 11 Inner Products, Norms and Orthogonality
5. 12 The Complex Exponential Fourier Series
6. 13 Convergence and Fourier’s Conjecture
7. 14 Convergence and Fourier’s Conjecture: The Proofs
8. 15 Derivatives and Integrals of Fourier Series
9. 16 Applications
10. III Classical Fourier Transforms
1. 17 Heuristic Derivation of the Classical Fourier Transform
2. 18 Integrals on Infinite Intervals
3. 19 The Fourier Integral Transforms
4. 20 Classical Fourier Transforms and Classically Transformable Functions
5. 21 Some Elementary Identities: Translation, Scaling and Conjugation
6. 22 Differentiation and Fourier Transforms
7. 23 Gaussians and Gaussian-Like Functions
8. 24 Convolution and Transforms of Products
9. 25 Correlation, Square-Integrable Functions and the Fundamental Identity
10. 26 Generalizing the Classical Theory: A Naive Approach
11. 27 Fourier Analysis in the Analysis of Systems
12. 28 Multi-Dimensional Fourier Transforms
13. 29 Identity Sequences
14. 30 Gaussians as Test Functions and Proofs of Important Theorems
11. IV Generalized Functions and Fourier Transforms
1. 31 A Starting Point for the Generalized Theory
2. 32 Gaussian Test Functions
3. 33 Generalized Functions
4. 34 Sequences and Series of Generalized Functions
5. 35 Basic Transforms of Generalized Fourier Analysis
6. 36 Generalized Products, Convolutions and Definite Integrals
7. 37 Periodic Functions and Regular Arrays
8. 38 Pole Functions and General Solutions to Simple Equations
12. V The Discrete Theory
1. 39 Periodic, Regular Arrays
2. 40 Sampling, Discrete Fourier Transforms and FFTs
14. Index

## Product information

• Title: Principles of Fourier Analysis, 2nd Edition
• Author(s): Kenneth B. Howell
• Release date: December 2016
• Publisher(s): CRC Press
• ISBN: 9781498734134